Article
Ufa Mathematical Journal
Volume 11, Number 4, pp. 115-130
Realization of homogeneous v Triebel-Lizorkin spaces with $p=\infty$ and characterizations via differences
Benallia M., Moussai M.
DOI:10.13108/2019-11-4-115
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In this paper, via the decomposition of Littlewood-Paley,
the homogeneous Triebel-Lizorkin space $\dot{F}_{\infty,q}^{s}$ is defined on $\mathbb{R}$ by distributions modulo polynomials in the sense that $\|f\|=0$ ($\|\cdot\|$ the quasi-seminorm in $\dot F^{s}_{\infty,q}$) if and only if $f$ is a polynomial on $\mathbb{R}$. We consider this space as a set of ``true'' distributions and we are lead to examine the convergence of the Littlewood-Paley sequence of each element in $\dot F^{s}_{\infty,q}$. First we use the realizations and then we obtain the realized space $\dot{\widetilde{F}}{^{s}_{\infty,q}}$ of $\dot{F}_{\infty,q}^{s}$.
Our approach is as follows. We first study the commuting translations and dilations of realizations in $\dot{F}_{\infty,q}^{s}$, and employing distributions vanishing at infinity in the weak sense, we construct $\dot{\widetilde{F}}{^{s}_{\infty,q}}$. Then, as
another possible definition of $\dot{F}_{\infty,q}^{s}$, in the case $s>0$, we make use of the differences and
describe $\dot{\widetilde{F}}{^{s}_{\infty,q}}$ as $s>\max(n/q-n,0)$.